2.8. LINEAR TRANSFORMATION II
2.8
Linear Transformation II
MATH 294
2.8.1
71
SPRING 1987
PRELIM 3
#3
2
Consider the subspace of C∞
given by all things of the form
a sin t + b cos t
,
~x(t) =
c sin t + d cos t
where a,b,c & d are arbitrary constants. Find a matrix representation of the linear
transformation
T (~x) = D~x, where D~x ≡ ~x˙ .
carefully define any terms you need in order to make this representation. Hint: A
good basis for this vector space starts something like this
sin t
,... .
0
MATH 294
2.8.2
MATH 294
2.8.3
PRELIM 3
#5
spr97
FINAL
#2
2
2
which is given by T (x) = ẋ
to C∞
T is linear transformation from C∞
MATH 294
2.8.4
SPRING 1987
The idea of eigenvalue λ and eigenvector v can be generalized from matrices and
<n to linear transformations and their related vector spaces. If T (v) = λv (and
v 6= 0) then λ is an eigenvalue of T , and v is its associated eigenvector.
1
with x(0) =
For the subspace of x(t) in C∞
x(1) = 0 find an eigenvalue and
2 0
2
2
eigenvector of T (x) = D x, where D x ≡ ẍ −
x. What is the kernel of T ?
0 3
FALL 1987
PRELIM 3
# 14
Find the kernel of the linear transformation
0 x1
0
T (x(t)) ≡
−
x02
−4
1
0
x1
x2
2
2
into C∞
where T transforms C∞
MATH 294
2.8.5
FALL
 1997

#5
 PRELIM
 3
x
x+y
Define T  y  ≡  x − z  , which is a linear transformation of <3 into itself.
y+z
z
a) Is T 1-1?
b) Is T onto?
c) Is T an isomorphism?
Substantiate your answers.
72
CHAPTER 2. MORE LINEAR ALGEBRA
MATH 294
2.8.6
FALL 1987
FINAL
#1
T is a linear transformation of <3 into <2 such that


 
 
1
2
1
2
1
1






T −1 =
, T 1 =
, T 1 =
.
1
0
−1
2
0
1
a) Is T 1-1?
b) Determine the matrix of T relative to the standard bases in <3 and <2 .
MATH 294
2.8.7
MATH 294
2.8.8
FALL 1987
FINAL
# 7a
Consider the boundary-value problem
X 00 + λX = 0, 0 < x < π, X(0) = X(π) = 0, where λ is a given real number.
a) Is the set of all solutions of this problem a subspace of C∞ [0, π]? why?
b) Let W = set of all functions X(x) in Ci nf ty[0, π] such that X(0) = X(π) = 0.
Is T ≡ D2 − λ linear as a transformation of W into C∞ [0, π]? Why?
c) For what values of λ is Ker(T ) nontrivial?
d) Choose one of those values of λ and determine Ker(T )
FALL 1989
PRELIM 3
#3
Let W be the following subspace of <3 ,

   
  
3
2
1
1
W = Comb  0  ,  1  ,  1  ,  3  .
−3
0
−1
1

 
1
1
a) Show that  0  ,  1  is a basis for W .
−1
1
For b) and c) below, let T be the following linear transformation T : W → <3 ,,



 
w1
w1
1 0 −1
T  w2  =  0 0 0  w2 
w3
w3
0 0 0


w1
for those w =  w2  in <3 which belong to W .
w3
[You are allowed to use a) even if you did not solve it.]
b) What is the dimension of Range(T )? (Complete reasoning, please.
c) What is the dimension of Ker(T )? (Complete reasoning, please.

2.8. LINEAR TRANSFORMATION II
MATH 294
2.8.9
FALL 1989
FINAL
73
#7
Let T : <2 → <2 be the linear transformation given in the standard basis for <2 by
x
x+y
T
=
.
y
0
a) Find the matrix
of T in the standard
basis for <2
1
1
b) Show that β =
,
is also a basis for <2 .
1
2
In c) below, you may use the result of b) even if you did not show it.
c) Find the matrix of T in the basis β given in b). (I.e., in T : <2 → <2 both copies
of <2 have the basis β.
MATH 294
2.8.10
SPRING 1990?
PRELIM 2
#4
Let A be a linear transformation from a vector space V to another vector space U .
Let (~v1 , . . . , ~vn ) be a basis for V and let (~u1 , . . . , ~un ) be a basis for U .
Suppose it is known that
A(~v1 ) = 2u2
A(~v2 ) = 3u3
..
.
A(~vi ) = (i + 1)~ui+1
..
.
A(~vn−1 ) = n~un
and A(~vn ) = 0 ← zero vector in U .
Can you find A(~v ) in terms of the ~ui ’s where
~v = ~v1 + ~v2 + . . . + ~vn =
n
X
i=1
~vi
74
CHAPTER 2. MORE LINEAR ALGEBRA
MATH 294
2.8.11
FALL 1991
FINAL
#8
T/F
c) If T : V → W is a linear transformation, then the range of T is a subspace of V .
d) If the range of T : V → W is W , then T is 1-1.
e) If the null space of T : V → W is {0}, then T is 1-1.
f ) Every change of basis matrix is a product of elementary matrices.
g) If T : U → V and S : V → W are linear transformations, and S is not 1-1, then
ST : U → W is not 1-1.
h) If V is a vector space with an inner product, (,), if {w
~ 1, w
~ 2, . . . , w
~ n } is an orn
P
thonormal basis for V , and if ~v is a vector in V , then ~v =
(~v , w
~ i )w
~ i.
i=1
T : Vn → Vn is an isomorphism if and only if the matrix which represents T in
any basis is non-singular.
j) If S and T are linear transformations of Vn into Vn , and in a given basis, S
is represented by a matrix A, and T is represented by a matrix B, then ST is
represented by the matrix AB
-note- Matrices are not necessarily square.
i)
MATH 294
2.8.12
MATH 294
2.8.13
SPRING 1992
PRELIM 3
#5
The vector space
V3 has the standard basis S = (~e1 , ~e2 , ~e3 ) and the basis B =
2~e2 , − 21 ~e1 , ~e3 .
a) Find the change
 of basis matrices (B : S) and (S : B). If a vector ~v has the
1
representation  1  in the standard basis, find its representation β(~v ) in the B
1
basis.
b) A transformation T is defined as follows: T ~v = the reflection of ~v across the x − z
plane in the standard basis. (For reflection, in V2 the reflection of a~i + b~j across
the x axis would be a~i − b~j. Find a formula for T in the standard basis. Why is
T a linear transformation?
c) Find TB , the matrix of T in the B basis.
d) Interpret T geometrically in the B basis, i.e., describe TB in terms of rotations,
reflections, etc.
FALL 1992
FINAL
#6
Let C 2 (−∞, ∞) be the vector space of twice continuously differentiable functions
on −∞ < x < ∞ and C 0 (inf ty, ∞) be the vector space of continuous functions on
−∞ < x < ∞.
a) Show that the transformation L : C 2 (∞, ∞) → C 0 (−∞, ∞) defined by Ly =
∂2y
∂x2 − 4y is linear.
b) Find a basis for the null space of L. Note: You must show that the vectors you
choose are linearly independent.
2.8. LINEAR TRANSFORMATION II
MATH 293
2.8.14
SPRING 1995
75
FINAL
#2
Let P 3 be the vector space of polynomials of degree ≤ 3, and let L : P 3 → P 3 be
given by
L(p)(t) = t
∂2p
(t) + 2p(t).
∂t2
a) Show that L is a linear transformation.
b) Find the matrix of L in the basis 1, t, t2 , t3 .
c) Find a solution of the differential equation
t
∂2p
+ 2p(t) = t3 .
∂t2
Do you think that you have found the general solution?
MATH 293
2.8.15
MATH 294
2.8.16
SPRING 1995
FINAL
#3
Let V be the vector space of real 3 × 3 matrices.
a) Find a basis of V . What is the dimension of V ?
Now consider the transformation L : V → V given by L(A) = A + AT .
b) Show that L is a linear transformation.
c) Find a basis for the null space (kernel) of L.
SPRING 1997
FINAL
# 10
Let P2 be the vector space of polynomials of degree ≤ 2, equipped with the inner
product
Z 1
< p(t), q(t) >=
p(t)q(t)dt
−1
Let T : P2 → P2 be the transformation which sends the polynomial p(t) to the
polynomial
(1 − t2 )p00 (t) − 2tp0 (t) + 6p(t)
a)
b)
c)
d)
e)
Show that T is linear.
Verify that T (1) = 6 and T (t) = 4t. Find T (t2 ).
Find the matrix A of T with respect to the standard basis = 1, t, t2 for P2 .
Find the basis for N ul(A) and Col(A).
Use the Gram-Schmidt process to find an orthogonal basis B for P2 starting form
.
76
CHAPTER 2. MORE LINEAR ALGEBRA
MATH 294
2.8.17
FALL 1997
PRELIM 3
#5
Let T : <2 → <2 be the linear transformation that rotates every vector (starting at
the origin) by θ degrees in the counterclockwise direction. Consider the following
two bases for <2 :
1
0
B=
,
,
0
1
and
C=
cos α
sin α
− sin α
,
.
cos α
a) Find the matrix [T ]B of T in the standard basis B.
b) Find the matrix [T ]C of T in the basis C. Does [T ]C depend on the angle α?
MATH 294
2.8.18
MATH 294
2.8.19
FALL 1997
FINAL
#9
Consider the vector space V of 2 matrices. Define a transformation T : V → V by
T (A) = AT , where A is an element of V (that is, it is a 2 × 2 matrix), and AT is
the transpose of A.
a) Show that T is linear transformation.
The value λ is an eigenvalue for T , and ~v 6= 0 is the corresponding eigenvector, if
T (~v ) = λ~v . (Note: here ~v is a 2 × 2 matrix).
b) Find an eigenvalue of T (You need only find one, not all of them). (Hint: Search
for matrices A such that T (A) is a scalar multiple of A.)
c) Find an eigenvector for the particular eigenvalue that yl=ou found in part (b).
d) Let W be the complete eigenspace of T with the eigenvalue from part (b) above.
Find a basis for W . What is the dimension of W ?
SPRING 1998
FINAL
#6
Let T : P 2 → P 3 be the transformation that maps the second order polynomial
p(t) into (1 + 2t)p(t),
a) Calculate T (1), T (t), and T (t2 ).
b) Show that T is a linear transformation.
c) Write the components of T (1), T (t), T (t2 ) with respect to the basis C =
{1, t, t2 , 1 + t3 }.
d) Find the matrix of T relative to the bases B = {1, t, t2 } and C = {1, t, t2 , 1 + t3 }.
2.8. LINEAR TRANSFORMATION II
MATH 294
2.8.20
FALL 1998
77
PRELIM 3
Consider the following three vectors in
 

1
~y =  0  , ~u1 = 
1
#1
<3



1
1
1  , and~u2 =  −1  .
1
0
[Note: ~u1 and ~u2 are orthogonal.].
a) Find the orthogonal projection of ~y onto the subspace of <3 spanned by ~u1 and
~u2 .
b) What is the distance between ~y and span{~u1 , ~u2 }?
c) In terms of the standard basis for <3 , find the matrix of the linear transformation
that orthogonally projects vectors onto span{~u1 , ~u2 }.
MATH 294
2.8.21
MATH 293
2.8.22
FALL 1998
FINAL
#4
Here we consider the vector spaces P1 , P2 , and P3 (the spaces of polynomials of
degree 1,2 and 3).
a) Which of the following transformations are linear? (Justify your answer.)
i) T : P1 → P3 , T (p) ≡ t2 p(t) + p(0)
ii) T : P1 → P1 , T (p) ≡ p(t) + t
b) Consider the linear transformation T : P2 → P2 defined by T (a0 + a1 t + a2 t2 ) ≡
2
(−a1 + a2 ) + (−a
0 + a1 )t + (a2 )t. with respect to the standard basis of P2 , β =
0 −1 1
{1, t, t2 }, is A =  −1 1 0 . Note that an eigenvalue/eigenvector pair of
0
0 1
 
0
A is λ = 1, v =  1  . Find an eigenvaue/eigenvector (or eigenfunction) pair of
1
T . That is, find λ and g(t) in P2 such that T (g(t)) = λg(t).
c) Is the set of vectors in P2 {3+t, −2+t, 1+t2 } a basis of P2 ? (Justify your answer.)
SPRING 19?
PRELIM 2
#4
Let M be the transformation from P n to P n such that
M p(t) =
1
[p(t) + p(−t)](t real)
2
a) If n = 3 find the matrix of this transformation with respect to the basis
{1, t, t2 , t3 }.
b) Let N = I − M. What is N p(t) in terms of p(t)? Show that M 2 = M M = M ,
MN = MN = 0
MATH 294
2.8.23
FALL 1987
PRELIM 2
# 3 MAKE-UP
a) If A is an n × n matrix with rank(A) = r, then what is the dimension of the
vector space of all solutions of the system of linear equations A~x = ~0
b) What is the dimension of the kernel of the linear transformation from <n to <n
which has A for its matrix in the standard basis.
78
CHAPTER 2. MORE LINEAR ALGEBRA
MATH 294
2.8.24
MATH 294
2.8.25
PRELIM 2
# 14 MAKE-UP
FALL 1987
FINAL
# 6 MAKE-UP
Let T : <2 → <4 
be a 
linear transformation.


3
−1
 1 
 0 
2
3
−9




a) If T
=   and T
=
, what is T
?
7
0
−1
1 
26
2
0
1
0
b) What are T
and T
?
0
1
1
1
c) What is the matrix of T in the basis
,
for <2 , and the standard
1
−1
basis for <4 ?
MATH 294
2.8.26
FALL 1987
~
Show that if T : V → W is a linear transformation
from
n
o V to W , and kernel(T ) = 0,
then T is 1-1. (Recall: kernel(T ) = ~v ∈ V | T (~v ) = ~0 .)
SUMMER 1989
PRELIM 2
#1
a)
b) Find a basis for ker(L), where L is linear transformation from <4 to <3 defined
by


 x

1 2 −4 3  1 
x2 
L(~x) ≡  1 2 −2 2  
 x3 
2 4 −2 3
x4
c) What is the dimension
 of ker(L)?
1
d) Is the vector ~y =  1  in range(L)? (Justify your answer.) If so, find all vectors
2
~x in <4 which satisfy L(~x) = ~y
MATH 294
2.8.27
SUMMER 1989
PRELIM 2
#4
Let P be the linear transformation from <3 to <3 defined by

 

x
x
P  y  =  y .
z
0
a) Find a basis for ker(P ).
b) Find a basis for range(P ).


1
c) Find all vectors ~x in <3 such that P ~x =  2  .
 0 
1
d) Find all vectors ~x in <3 such that P ~x =  2  .
3
2.8. LINEAR TRANSFORMATION II
MATH 293
2.8.28
SPRING 1995
PRELIM 3
79
#4
Let Lθ : <2 → <2 be the linear transformation which represent orthogonal projection onto the line `θ forming angle θ with the x-axis.
a) Find the matrix T of Lθ (with respect to the standard basis of <2 ).
b) Is Lθ invertible. Explain your answer geometrically.
c) Find all the eigenvalues of T .
MATH 294
2.8.29
FALL 1998
PRELIM 2
#1
The unit square OBCD below gets mapped to the parallelogram OB 0 C 0 D0 (on the
x1 − x3 plane) by the linear transformation T : <2 → <3 shown.
Problems (b) - (e) below can be answered with or without use of the matrix A from
part (a).
a) Is this transformation one-to-one? For this and all other short answer questions
on this test, some explanation is needed.)
b) What is the null space of A?
c) What is the column space of A?
d) Is A invertible? (No need to find the inverse if it exists.)
80
CHAPTER 2. MORE LINEAR ALGEBRA
MATH 294
2.8.30
FALL ?
FINAL
# 1 MAKE-UP
Consider the homogeneous system of equations B~x = ~0, where


x1


 
 x2 
0
1 0 −3 1
0



~

3 0  , ~x = 
x
0 .
B =  2 −1 0
,
and
0
=
 3 
 x4 
2 −3 0
0 4
0
x5
a) Find a basis for the subspace W ⊂ <5 , where W = set of all solutions of B~x = ~0.
b) Is B 1-1 (as a transformation of <5 → <3 )? Why?
c) Is B : <5 → <3 onto? Why?
 
3
d) Is the set of all solutions of B~x =  0  a subspace of <5 ? Why?
0